Question 4 - A-Level Maths

A two person zero-sum game is represented by the pay-off matrix for player A shown above.

Verify that there is no stable solution to this game. Player A intends to make a random choice between options $\mathrm { P } , \mathrm { Q }$ and R , choosing option P with probability $p _ { 1 }$, option Q with probability $p _ { 2 }$ and option R with probability $p _ { 3 }$ Player A wants to find the optimal values of $p _ { 1 } , p _ { 2 }$ and $p _ { 3 }$ using the Simplex algorithm. Player A formulates the following linear programming problem for the game, writing the constraints as inequalities. $$\begin{aligned} & \text { Maximise } P = V
& \text { subject to } V \geqslant 3 p _ { 1 } - 4 p _ { 2 } + p _ { 3 }
&
& V \geqslant - 2 p _ { 1 } + 4 p _ { 2 } + 2 p _ { 3 }
& V \geqslant - 2 p _ { 2 } - p _ { 3 }
& p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1
& p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , V \geqslant 0 \end{aligned}$$
Correct the errors made by player A in the linear programming formulation of the game, giving reasons for your answer.
Write down an initial Simplex tableau for the corrected linear programming problem. The Simplex algorithm is used to solve the corrected linear programming problem. The optimal values are $p _ { 1 } = 0.6 , p _ { 2 } = 0$ and $p _ { 3 } = 0.4$
Calculate the value of the game to player A.
Determine the optimal strategy for player B, making your reasoning clear.